Target tracking systems employing Kalman estimators for predicting the position of moving targets are frequently used for purposes of controlling intercept missiles and aircraft. In a typical radar tracking system, pulses are transmitted through an antenna at a predetermined repetition rate toward a target and the pulses are reflected from the target back to the antenna. The time of reception and the doppler shift of the pulses, together with the pointing angles of the airborne antenna, the time history of angular orientation and of the velocity vector of the skin tracking aircraft or missile are processed by a signal processor to generate signals that represent range, radial velocity or range rate, and the elevation and azimuth angles to the target.
In the mechanization of such a system, a high-speed digital computer may be used which operates on the measured input signals within a specified time frame. Calculations are made in accordance with the computer algorithm and the results of each calculation is sent to the antenna for controlling the antenna position to track the target.
From the input information, estimates or predictions of target position are generated at predetermined rates. The target position estimation signals are calculated from the last estimated position, target velocity and target acceleration estimation signals, and are utilized to point the antenna at the moving target and to make adjustments in the flight path of the missile. An optimal estimating system that is well suited for program implementation in a high-speed digital computer is the estimator known as a Kalman filter. The Kalman filter is well known in the literature and may be defined as an optimal recursive filter that is based on space and time domain formulations.
Typically, a Kalman filter or estimator processes the measured information concerning moving targets such as range, radial velocity, elevation and azimuth to develop signals that represent estimates of target relative position, target relative velocity and target acceleration. An additional set of parameters is developed representing the uncertainty in the estimation of target position and its time derivatives. The elements of this set of parameters are called the error covariances of the estimation model. A second set of error covariances represents the mean squared error in measurement of range, radial velocity, azimuth and elevation.
Any difference between a predicted value of an estimated quantity and its measured value is commonly called a residual. This residual is composed of errors in estimation and errors in measurement. Clearly, not all of an observed residual should be used to correct errors in estimation since the residual itself contains measurement errors. A Kalman gain factor is formulated which seeks to take that fraction of a residual which is due to estimation error alone. This fraction of the residual is then used to revise the estimation model after each observation or measurement. The revised estimates are then used to predict the results of the next measurement, and the process is repeated.
The measured quantities as well as the quantities for predicting the position of the target must be referenced to a coordinate system. Typically, a Cartesian coordinate system in the inertial reference is employed for simplicity reasons. A line-of-sight (LOS) or antenna coordinate system which extends along three axes, or alternatively, an aircraft or missile coordinate system may be used, with the longitudinal axis of the aircraft or missile being the basis for a three-axis system. In addition, an onboard inertial reference unit (IRU) supplies information as to the missile state and position in the inertial frame.
The signals described above, as well as tracking error signals of the antenna, are input to and operated upon by the onboard executive computer to calculate the various output signals for positioning the antenna to maintain its track on a target and to control the missile itself. These signals are employed to formulate a liner dynamic model to provide predictions of target position, velocity and acceleration. Measured quantities, such as range, range rate, elevation and azimuth angles and interdependent when calculating target position, velocity and acceleration. For n interdependent parameters there would be n.times.n sets of calculations involved in the direct generation of the Kalman gain factors. For the three spatial components (the LOS axes mentioned above) of target position, velocity and acceleration, n=9 in a stable, for example, geographic, coordinate system.
In a LOS coordinate system, the measured quantities of range, range rate, azimuth and elevation angles are independent of each other. When using a LOS coordinate system, the Kalman gain computations are greatly simplified and the number of computations are substantially reduced. However, the orientation of the LOS system moves with time as the antenna-carrying aircraft or missile moves in three-dimensional space. In conventional systems, formulated wholly within the LOS coordinate system, this change in the LOS orientation customarily employs rate gyros to measure the reorientation and results in a non-linear system model to predict the target's position, velocity and acceleration. Nonlinear system models require more complex computations involving complicated weighting factors to make these predictions.
It should be noted that the above discussion assumes that measured range information is available as an input to the Kalman filter. When range or range rate information is not available, the problems associated with controlling the missile flight toward intercept of the target are significantly increased. In a jamming environment accurate measures of target range and range rate information are effectively denied.
For target state estimation in a jamming environment, only the passive LOS data from sensors onboard the missile are generally available for midcourse guidance. A straightforward triangulation method that makes use of target LOS from both the missile and the mother ship, as well as the IRU supplied missile position relative to the launching platform, can be used to estimate the target location. This deterministic scheme relies exclusively on the latest fix, or position constraints, which tend to forfeit all information extrapolated from previous data and kinematic history. This can cause the system to be vulnerable to occasional large errors in the low resolution angular data or data dropouts from uplink.
Another possible approach is to use a weighted least squares filter for target ranging with some simple target modeling assumed over a finite filter memory length. Major drawbacks here are the inflexibility due to the batch-processing nature of the filter and the insufficiency in modeling the missile IRU error contribution.
A recursive, Kalman-type, digital, optimal filtering technique for a complete model of the target, missile and measurements offers considerable improvement in accuracy and ease of implementation over the weighted least squares filtering method. However, the optimal Kalman filtering approach to the problem involves not only the modeling of the target state, but also the missile state, IRU errors, measurement biases, and other systematic errors. This requires an eighteenth order filter and imposes an unacceptable computer burden on the available on-line estimation scheme. A module decoupling that estimates only the target acceleration, velocity and position in the downrange, off-range and altitude components, will reduce the filter to the order of nine. Each filter iteration would take about 24 ms to process the first set of sensor input data following the extrapolation, and processing each additional set of input data from other sensors adds about 3 ms. Including models for the two IRU misalignment angle errors increases these estimates to 43 and 4 ms, respectively. Thus, implementing the three-dimensional estimator with IRU correction is marginal with present computer speed and system frame of about 100 ms.
In addition to the actual implementation limitation problem, the higher order filter imposes more severe requirements on component tolerances such as unmodeled IRU error than a lower order scheme. If the component tolerance can be met, the higher order scheme should render more accurate estimates, but as the uncertainty increases, the performance will degrade much faster than for the lower order state-reduction system.
Finally, in a multi-mode passive ranging guidance system where target data is being directly received by missile onboard sensors and indirectly from the mother ship uplinks, IRU errors result in missile platform tilt and alignment errors in the missile-to-target LOS that must be corrected in order to obtain accurate LOS angle estimates.